1000 Lockers and 1000 Students!

A school has 1000 lockers and 1000 students on the first morning of term, all the lockers are open.
Everystudent walks in one by one.
The first student walks in and closes every locker
student number 2 opens every second locker
the third student changes the state of every third locker(opens the closed ones and closes the open ones)
4th student changes every 4th locker... and so on

Once all 1000 students have completed this process, which lockers are closed and why?

First of all lets take a look at what it takes for a locker to be closed or open.  

For a locker to be open we need it to be changed an even number of times because all the lockers started open.  For a locker to be closed we need it to be changed an odd number of times.

So if we changed a locker an even amount of times, lets say 4 times, then it would 

open(because every locker starts open)

closed(1st time changed)

open(2nd time changed)

closed(3rd time changed)

open(4th time changed)

If we changed a locker say 3 times it should be clear that it would end up closed.

So now we need to figure out which lockers will be changed an even amount of times vs the lockers that will be changed a odd amount of times.

Lets take a look at a couple of examples to try and tackle this

Take locker 10 for example.

We know that locker 10 started open (because every locker started open), then it was closed because student #1 closed it, then it was open because student #2, then closed because of student #5, and then finally opened because of student #10.  Each locker is changed by the number of factors(he numbers it is divisible by) it has.  So 10 has factors (1,2,5,10).  Because this is an even amount of factors, in our case there is 4, we know that the locker will end up open.

Now take locker 16 for example.  We know its factors are (1,2,4,8,16).  This is an odd number so we know the locker will end up closed.

How do we know which numbers have an even amount of factors? And which ones have an odd amount?

To answer that we need to look at how factors work.  

Every number has factors that are paired up.  So for example 10 has (10x1,1x10,2x5,5x2).  These pairings produce an even number of factors.  Can you think of a case in which this isn't always true?  If you are having trouble, take a look at the number 16 for example.  16 is a perfect square so we have all the pairs plus the square itself. So 16 has (1x16,16x1,8x2,2x8) and (4x4).  In all the other cases the factors are paired up, producing an even number.  With perfect squares we get an odd number because the factors are paired up except for the square itself.  Still don't get it?  The image below may help you understand it a bit better!

So in conclusion all the numbers that are perfect squares will be closed so 1,4,9,16,25......961.  

and every non perfect square will be open.

Some teacherly notes:

If students are having trouble I think it would be a good idea to pair them up or set them into groups.  This task seems better suited for teamwork!

To challenge your students you can ask them other relevant questions such as
How many times does locker x get changed / touched?
Which locker gets changed the most?
Which locker/s are touched exactly 4 times? etc.